Complexity Classification of Some Edge Modification Problems Assaf Natanzon , Ron Shamir , and Roded Sharan Department of Computer Science, Tel Aviv University, Tel-Aviv, Israel. g fnatanzon,shamir,roded @math.tau.ac.il. Abstract. In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove in this paper the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be char- acterized by a finite set of forbidden induced subgraphs. 1 Introduction Problem Definition: Edge modification problems call for making small changes to the edge set of an input graph in order to obtain a graph with a desired property. These include completion, deletion and editing problems. Let be a family of G V E graphs. In the -Editing problem the input is a graph , and the goal V V G V EF is to find a minimum set F such that , where F E F E denotes the symmetric difference between and .Inthe -Deletion E problem only edge deletions are permitted, i.e., F . The problem is equiva- lent to finding a maximum subgraph of G with property .Inthe -Completion E problem one is only allowed to add edges, i.e., F . Equivalently, we seek a minimum supergraph of G with property . In this paper we study edge modification problems with respect to some well-studied graph properties. Motivation: Graph modification problems are fundamental in graph theory. Al- ready in 1979, Garey and Johnson mentioned 18 different types of vertex and edge modification problems [11, Section A1.2]. Edge modification problems have applications in several fields, including molecular biology and numerical algebra. In many application areas a graph is used to model experimental data, and then edge modifications correspond to correcting errors in the data: Adding an edge corrects a false negative error, and deleting an edge corrects a false positive error. We summarize below some of these applications. Definitions of the graph classes are given in Section 3. Interval modification problems have important applications in physical mapping of DNA (see [5, 8, 12, 14]). Depending on the biotechnology used and the kind of experimental errors, completion, deletion and editing problem arise, both for interval graphs and for unit interval graphs. The chordal completion problem, which is also called the minimum fill-in prob- lem, arises when numerically performing a Gaussian elimination on a sparse sym- metric positive-definite matrix [30]. Chordal deletion problems occur when trying to solve the CLIQUE problem. Some heuristics for finding a large clique (see, e.g., [33]) aim to find a maximum chordal subgraph of the input graph, on which a maximum clique can be found in polynomial time. Previous Results: Strong negative results are known for vertex deletion prob- lems: Lewis and Yannakakis [24] showed that for any property which is non- trivial and hereditary, the maximum induced subgraph problem is NP-complete. Furthermore Lund and Yannakakis[26] proved that for any such property, and for every , the maximum induced subgraph problem cannot be approximated log n P NP with ratio in quasi-polynomial time, unless . (Throughout m we use n and to denote the number of vertices and edges, respectively, in a graph). For edge modification problems no such general results are known, although some attempts have been made to go beyond specific graph properties [3, 10, 2]. Most of the results obtained so far concerning edge modification problems are NP-hardness ones. (For simplicity we shall often refer to the decision ver- sion of the optimization problems). Chain Completion and Chordal Completion were shown to be NP-complete in [34]. As noted in [12], the NP-completeness of Interval Completion and Unit Interval Completion also follows from [34]. In- terval Completion was directly shown to be NP-complete in [11, problem GT35] and [23]. Deletion problems on interval graphs and unit interval graphs were proven to be NP-complete in [12]. Cograph Completion and Cograph Deletion were shown to be NP-complete in [10]. Threshold Completion and Threshold Deletion were shown to be NP-complete in [27]. Comparability Completion was shown to be NP-complete in [17] and Comparability Deletion was shown to be NP-complete in [35]. Much fewer results are known for editing problems: Chordal Editing was proven to be NP-complete in [4]. The connected bipartite interval (caterpillar) editing problem was proven to be NP-complete in [8]. Split Editing was shown to be polynomial in [19]. Several authors studied variants of the completion problem, motivated by DNA mapping, in which the input graph is pre-colored and the required supergraph also obeys the coloring (see [5] and references thereof). Other biologically moti- vated problems, called sandwich problems, seek a supergraph satisfying a given property which does not include (pre-defined) forbidden edges. Polynomial al- gorithms or NP-hardness results are known for many sandwich problems [16, 15, 18, 21]. Several results on the parametric complexity of completion problems were also obtained [22, 7]. k Approximation algorithms exist for several problems. In [28] an approxima- tion algorithm is given for the minimum fill-in problem, where k denotes the size m log n of an optimum solution. In [1] an O approximation algorithm is given for the minimum chordal supergraph problem (where one wishes to mini- mize the total number of edges in the resulting graph) For the minimum interval log n supergraph problem an O approximation algorithm was given in [29]. In [8] it was shown that the minimum number of edge editions needed in order to convert a graph into a caterpillar cannot be approximated in polynomial time to n within an additive term of O , for , unless P=NP. Contribution of this paper: In this paper we study the complexity of edge modi- fication problems on some well-studied classes of graphs. We show, among other results, that deletion problems are NP-hard for perfect, chain, chordal, split and asteroidal triple free graphs; and that editing problems are NP-hard for perfect and comparability graphs. We also show that it is NP-hard to approximate compara- bility modification problems to within a factor of . The reader is referred to Figure 1 which summarizes the complexity results for the (decision version of) modification problems that we considered. Positive complexity results are given for bounded degree input graphs: We give a simple, general constant factor approximation algorithm for the deletion and editing problems w.r.t. any hereditary property that is characterized by a finite set of forbidden induced subgraphs. We also show that Chain Deletion and Editing, Split Deletion and Threshold Deletion and Editing become polynomial when the input degrees are bounded. Organization of the paper: Section 2 contains simple basic results that show connections between the complexity of related modification problems. Section 3 contains the main hardness results. Section 4 gives the positive results on bounded degree graphs. For lack of space, some proofs are omitted and many corollaries are only alluded to in Figure 1. 2 Basic Results In this section we summarize some easy observations on modification problems, which will help us deduce complexity results from results on related graph fami- lies, and concentrate on those modification problems which are meaningful. Definitions and Notation: All graphs in this paper are simple and contain no V E V self-loops. Let G be a graph. We denote its set of vertices also by G G G G V E V . We denote by the complement graph of , i.e., , where V V n E E . (Throughout, we abuse notation for the sake of brevity, S S S fs s s s S s s g and for a set we use to denote .) If U V E G is bipartite then its bipartite complement is the bipartite graph G U V E A V U V n E , where E . For a subset we denote by G A v V A the subgraph induced on the vertices of . For a vertex we denote v v G v V by N the set of vertices adjacent to in . For a vertex we denote by v v G G the graph obtained by adding to as an isolated vertex. We denote by v G v G the graph obtained from by adding and connecting it to every other G G vertex of G. For a graph property the notation implies that satisfied . For basic definitions of graph properties and much more on the graph classes discussed here see, e.g., [13, 6]. F G V E F Let be a graph property. If is a set of non-edges such that F j k F k k and j , then is called a -completion set w.r.t. -deletion set and k -editing set are similarly defined. G Basic Results: A graph property is called hereditary if when a graph sat- G isfies every induced subgraph of satisfies . is called hereditary on G subgraphs if when G satisfies , every subgraph of satisfies . is called G ancestral if when G satisfies , every supergraph of satisfies . Proposition 1. If property is hereditary on subgraphs then -Deletion and -Editing are polynomially equivalent, and -Completion is not meaningful. Proposition 2. If is an ancestral graph property then -Completion and - Editing are polynomially equivalent, and -Deletion is not meaningful.